Hi Mats,
If I have this right, I think you are looking for some help with modeling continuous zero inflated data? I believe some options for that are beta and gamma regression.
In your shoes I might try examining your outcome distribution with fitdistrplus to get a finer sense of the shape of the distribution (of course still directed mainly by your domain knowledge of the process) to see what may be a good fit (For instance, I'm not sure I would immediately describe the plot you included as log normal...). I am a big fan of the Cullen and Frey plot to give you an idea of the distribution family.
You can then start simple with say a glm with family = gamma, if that happens to be appropriate, or take a look at some more specialized packages in appropriate/needed for your data. Hope that helps!
Cheers,
Ben
Some possible resources that might help: